Answer
See explanation
Work Step by Step
Method: Proof by contradiction
Hypothesis:
Let $g$ and $h$ be two distinct inverses of the function $f(x)$ and $I_x$ be the identity function in x.
Proof:
So, $(g∘f)(x)=(f∘g)(x)=I_x$
And, $(h∘f)(x)=(f∘h)(x)=I_x$
Therefore,
$(f∘g)(x)=(f∘h)(x)$
Composing by left side,
$g∘(f∘g)=g∘(f∘h)$
By associativity, we can say that
$(g∘f)∘g=(g∘f)∘h$
As $(g∘f)(x)=I_x$.
$I_x∘g=I_xoh$
which means that
$g(x)=h(x)$
But we assumed they were distinct. Thus our hypothesis isn't self-consistent and leads to contradiction and hence it is proven to be false.
